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In this paper, we study a tumor growth equation along with various models for the nutrient component, including a *in vitro* model and a *in vivo* model. At the cell density level, the spatial availability of the tumor density $ n$ is governed by the Darcy law via the pressure $ p(n) = n^{γ}$. For finite $ γ$, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As $ γ \to ∞$, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.

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